Sequences & series

What is the position of the term that you want to find? The position is the value of '$n$' that we substitute into the general formula.

To find the missing term, we use the equation for the general term:

For the missing term $c$, we have $n = 4$:

To find the two missing terms, we use the equation for the general term:

For the missing term $q$, we have $n = 2$:

For the missing term $y$, we have $n = 4$:

What is the position of the term that you want to find? The position is the value of '$n$' that we substitute into the general formula.

To find the missing term, we use the equation for the general term:

For the missing term $b$, we have $n = 4$:

To find the two missing terms, we use the equation for the general term:

For the missing term $q$, we have $n = 2$:

For the missing term $z$, we have $n = 4$:

What is the position of the term that you want to find? The position is the value of '$n$' that we substitute into the general formula.

To find the missing term, we use the equation for the general term:

For the missing term $b$, we have $n = 4$:

To find the two missing terms, we use the equation for the general term:

For the missing term $t$, we have $n = 2$:

For the missing term $y$, we have $n = 4$:

Calculate the common difference:

To complete the sequence, we need to determine the common difference:

To complete the sequence, we need to determine the common difference:

To complete the sequence, we need to determine the common difference in terms of $v$:

Calculate the common difference:

To complete the sequence, we need to determine the common difference:

To complete the sequence, we need to determine the common difference:

To complete the sequence, we need to determine the common difference in terms of $v$:

Calculate the common difference:

To complete the sequence, we need to determine the common difference:

To complete the sequence, we need to determine the common difference:

To complete the sequence, we need to determine the common difference in terms of $c$:

The question gives us $\text{4}$ terms of a sequence and specifically tells us that the terms follow an arithmetic pattern. We must find the common difference of the sequence, $d$.

This is an example of a non-numeric arithmetic sequence, where each term in the given sequence is an algebraic expression. To find the common difference ($d$), we subtract any two successive terms in the sequence:

We could also have chosen the following terms:

As expected, both calculations give the same value for $d$.

Therefore, the common difference between each term in the sequence is $2 b$.

We have the common difference of the sequence and the first $\text{4}$ terms, and the question asks us to find the next $\text{2}$ terms.

To determine any term in the sequence, we add the common difference to the term before it. In other words: $T_{n} = T_{n - 1} + d$. Thus to find the fifth term, add the common difference to the fourth term.

Repeat the calculation to continue to the next term:

Therefore, the required terms are: $T_{5} = 11 b - 5 m$ and $T_{6} = 13 b - 5 m$.

The question gives us $\text{4}$ terms of a sequence and specifically tells us that the terms follow an arithmetic pattern. We must find the common difference of the sequence, $d$.

This is an example of a non-numeric arithmetic sequence, where each term in the given sequence is an algebraic expression. To find the common difference ($d$), we subtract any two successive terms in the sequence:

We could also have chosen the following terms:

As expected, both calculations give the same value for $d$.

Therefore, the common difference between each term in the sequence is $- 2 u$.

We have the common difference of the sequence and the first $\text{4}$ terms, and the question asks us to find the next $\text{2}$ terms.

To determine any term in the sequence, we add the common difference to the term before it. In other words: $T_{n} = T_{n - 1} + d$. Thus to find the fifth term, add the common difference to the fourth term.

Repeat the calculation to continue to the next term:

Therefore, the required terms are: $T_{5} = 3 h + u$ and $T_{6} = 3 h - u$.

The question gives us $\text{3}$ terms of a sequence and specifically tells us that the terms follow an arithmetic pattern. We must find the common difference of the sequence, $d$.

This is an example of a non-numeric arithmetic sequence, where each term in the given sequence is an algebraic expression. To find the common difference ($d$), we subtract any two successive terms in the sequence:

We could also have chosen the following terms:

As expected, both calculations give the same value for $d$.

Therefore, the common difference between each term in the sequence is $- 3 q$.

We have the common difference of the sequence and the first $\text{3}$ terms, and the question asks us to find the next $\text{2}$ terms.

To determine any term in the sequence, we add the common difference to the term before it. In other words: $T_{n} = T_{n - 1} + d$. Thus to find the fourth term, add the common difference to the third term.

Repeat the calculation to continue to the next term:

Therefore, the required terms are: $T_{4} = - 19 q + s$ and $T_{5} = - 22 q + s$.

To find the common difference, we subtract any two successive terms in the sequence. Let us use the third term $\left(T_{3}\right)$ and the fourth term $\left(T_{4}\right)$. (We could also have chosen the second term $\left(T_{2}\right)$ and the third term $\left(T_{3}\right)$, both should give us the same answer.)

We find the difference between any term in the sequence and the term before it:

We can check that $d$ is constant by subtracting two other terms in the sequence:

Therefore, the common difference is $\text{7}$.

Add the common difference to the last known term to get the next term in the sequence:

Using the same method we can calculate the following two terms in the sequence:

Therefore the sequence is,

To find the common difference, we subtract any two successive terms in the sequence. Let us use the second term $\left(T_{2}\right)$ and the third term $\left(T_{3}\right)$. (We could also have chosen the first term $\left(T_{1}\right)$ and the second term $\left(T_{2}\right)$, both should give us the same answer.)

We find the difference between any term in the sequence and the term before it:

We can check that $d$ is constant by subtracting two other terms in the sequence:

Therefore, the common difference is $-\text{2}$.

Add the common difference to the last known term to get the next term in the sequence:

Using the same method we can calculate the following three terms in the sequence:

Therefore the sequence is,

To find the common difference, we subtract any two successive terms in the sequence. Let us use the first term $\left(T_{1}\right)$ and the second term $\left(T_{2}\right)$. (We could also have chosen the second term $\left(T_{2}\right)$ and the third term $\left(T_{3}\right)$, both should give us the same answer.)

We find the difference between any term in the sequence and the term before it:

We can check that $d$ is constant by subtracting two other terms in the sequence:

Therefore, the common difference is $\text{13}$.

Add the common difference to the last known term to get the next term in the sequence:

Using the same method we can calculate the following three terms in the sequence:

Therefore the sequence is,

The key to starting this question is the fact that the terms form an arithmetic sequence: that means that the difference between the first and second terms is equal to the difference between the second and third terms.

The given sequence is arithmetic, so the difference between successive terms will be the same (common). So we know that $T_{2} - T_{1} = T_{3} - T_{2}$ and we can solve for $z$:

For an equation with fractions, we can multiply both sides of the equation by the LCD of the fractions to remove the denominators. In this case, the LCD is $\text{3}$.

Solve for $z$:

Therefore, $z = -3$.

The first three terms in the sequence are given in terms of $z$. We have calculated the value of $z$, so we can substitute and simplify to find the terms.

To find the values of the terms, we need to substitute $z = -3$ into the expressions for each of the three terms:

Therefore, the first three terms of the sequence are: $-3 , -7$ and $-11$.

The key to starting this question is the fact that the terms form an arithmetic sequence: that means that the difference between the first and second terms is equal to the difference between the second and third terms.

The given sequence is arithmetic, so the difference between successive terms will be the same (common). So we know that $T_{2} - T_{1} = T_{3} - T_{2}$ and we can solve for $y$:

For an equation with fractions, we can multiply both sides of the equation by the LCD of the fractions to remove the denominators. In this case, the LCD is $\text{2}$.

Solve for $y$:

Therefore, $y = -3$.

The first three terms in the sequence are given in terms of $y$. We have calculated the value of $y$, so we can substitute and simplify to find the terms.

To find the values of the terms, we need to substitute $y = -3$ into the expressions for each of the three terms:

Therefore, the first three terms of the sequence are: $- \dfrac{11}{2} , \dfrac{5}{2}$ and $\dfrac{21}{2}$.

The key to starting this question is the fact that the terms form an arithmetic sequence: that means that the difference between the first and second terms is equal to the difference between the second and third terms.

The given sequence is arithmetic, so the difference between successive terms will be the same (common). So we know that $T_{2} - T_{1} = T_{3} - T_{2}$ and we can solve for $p$:

For an equation with fractions, we can multiply both sides of the equation by the LCD of the fractions to remove the denominators. In this case, the LCD is $\text{3}$.

Solve for $p$:

Therefore, $p = 2$.

The first three terms in the sequence are given in terms of $p$. We have calculated the value of $p$, so we can substitute and simplify to find the terms.

To find the values of the terms, we need to substitute $p = 2$ into the expressions for each of the three terms:

Therefore, the first three terms of the sequence are: $- \dfrac{14}{3} , - \dfrac{17}{3}$ and $- \dfrac{20}{3}$.

The question gives us 4 terms of a sequence and specifically tells us that the numbers form an arithmetic pattern. We need to find the common difference of that pattern, $d$.

To determine the common difference for an arithmetic sequence, we subtract any two successive terms in the sequence: $d = T_{n} - T_{n - 1}$. In other words the common difference is the difference between any term in the sequence and the term before it:

Or we could also have chosen:

Notice that both calculations give the same value for $d$.

Therefore the common difference is $d = - \frac{4}{5}$.

Now we must find the next $\text{2}$ terms in the sequence. We can do this using the common difference: any term in the sequence is equal to the term before it plus the common difference: $T_{n} = T_{n - 1} + d$.

To calculate the fifth term, add the common difference to the fourth term:

Continue the sequence by adding $d = - \frac{4}{5}$ again:

Therefore, the next two terms in the sequence are: $- \frac{66}{5}$ and $-14$.

The full sequence is $-10 \; ; \; - \frac{54}{5} \; ; \; - \frac{58}{5} \; ; \; - \frac{62}{5} \; ; \; - \frac{66}{5} \; ; \; -14 \; ; - \frac{74}{5} \; ; \; \ldots$.

The question gives us 6 terms of a sequence and specifically tells us that the numbers form an arithmetic pattern. We need to find the common difference of that pattern, $d$.

To determine the common difference for an arithmetic sequence, we subtract any two successive terms in the sequence: $d = T_{n} - T_{n - 1}$. In other words the common difference is the difference between any term in the sequence and the term before it:

Or we could also have chosen:

Notice that both calculations give the same value for $d$.

Therefore the common difference is $d = - \frac{3}{2}$.

Now we must find the next $\text{2}$ terms in the sequence. We can do this using the common difference: any term in the sequence is equal to the term before it plus the common difference: $T_{n} = T_{n - 1} + d$.

To calculate the seventh term, add the common difference to the sixth term:

Continue the sequence by adding $d = - \frac{3}{2}$ again:

Therefore, the next two terms in the sequence are: $-2$ and $- \frac{7}{2}$.

The full sequence is $7 \; ; \; \frac{11}{2} \; ; \; 4 \; ; \; \frac{5}{2} \; ; \; 1 \; ; \; - \frac{1}{2} \; ; -2 \; ; \; - \frac{7}{2} \; ; \; -5 \; ; \; \ldots$.

The question gives us 4 terms of a sequence and specifically tells us that the numbers form an arithmetic pattern. We need to find the common difference of that pattern, $d$.

To determine the common difference for an arithmetic sequence, we subtract any two successive terms in the sequence: $d = T_{n} - T_{n - 1}$. In other words the common difference is the difference between any term in the sequence and the term before it:

Or we could also have chosen:

Notice that both calculations give the same value for $d$.

Therefore the common difference is $d = - \frac{3}{5}$.

Now we must find the next $\text{2}$ terms in the sequence. We can do this using the common difference: any term in the sequence is equal to the term before it plus the common difference: $T_{n} = T_{n - 1} + d$.

To calculate the fifth term, add the common difference to the fourth term:

Continue the sequence by adding $d = - \frac{3}{5}$ again:

Therefore, the next two terms in the sequence are: $\frac{18}{5}$ and $3$.

The full sequence is $6 \; ; \; \frac{27}{5} \; ; \; \frac{24}{5} \; ; \; \frac{21}{5} \; ; \; \frac{18}{5} \; ; \; 3 \; ; \frac{12}{5} \; ; \; \ldots$.

The common difference for terms in an arithmetic pattern: $d = T_{n+1} - T_{n}$.

We need to calculate the common difference $d$ at least two different ways to make sure that the values are the same:

Each calculation gives the same answer, which means we have found the common difference: $d = - 3 p + 2$.

Substitute $p = 1$ into the expressions for $T_{1}$ and $T_{2}$:

Therefore, $T_{1}$ = -5 and $T_{2}$ = -6.

The common difference for terms in an arithmetic pattern: $d = T_{n+1} - T_{n}$.

We need to calculate the common difference $d$ at least two different ways to make sure that the values are the same:

Each calculation gives the same answer, which means we have found the common difference: $d = 3 y + 1$.

Substitute $y = -1$ into the expressions for $T_{1}$ and $T_{2}$:

Therefore, $T_{1}$ = 3 and $T_{2}$ = 1.

The common difference for terms in an arithmetic pattern: $d = T_{n+1} - T_{n}$.

We need to calculate the common difference $d$ at least two different ways to make sure that the values are the same:

Each calculation gives the same answer, which means we have found the common difference: $d = z - 2$.

Substitute $z = -3$ into the expressions for $T_{1}$ and $T_{2}$:

Therefore, $T_{1}$ = 10 and $T_{2}$ = 5.

Calculate the common difference:

Determine the general formula:

Find the three terms:

Therefore, $\; T_{10} = \text{26}$, $\; T_{15} = \text{46} \;$ and $\; T_{30} = \text{106}$.

Find the common difference:

Now determine the general formula:

Finally, calculate the three required terms:

Therefore, $\; T_{10} = \text{56}$, $\; T_{15} = \text{81} \;$ and $\; T_{30} = \text{156}$.

Calculate the common difference:

Determine the general formula:

Find the three terms:

Therefore, $\; T_{10} = -\text{10}$, $\; T_{15} = -\text{20} \;$ and $\; T_{30} = -\text{50}$.

Find the common difference:

Now determine the general formula:

Finally, calculate the three required terms:

Therefore, $\; T_{10} = \text{105}$, $\; T_{15} = \text{150} \;$ and $\; T_{30} = \text{285}$.

Calculate the common difference:

Determine the general formula:

Find the three terms:

Therefore, $\; T_{10} = -\text{25}$, $\; T_{15} = -\text{35} \;$ and $\; T_{30} = -\text{65}$.

Find the common difference:

Now determine the general formula:

Finally, calculate the three required terms:

Therefore, $\; T_{10} = \text{76}$, $\; T_{15} = \text{116} \;$ and $\; T_{30} = \text{236}$.

First find the values of $a$ and $d$ for this series.

The formula for the general term of an arithmetic series is

$a$ is the first term of the series. So here $a = -3$.

$d$ is the constant difference between the terms of the series. We can determine this by finding the difference between any two consecutive terms of the series. Taking the difference of the first two terms:

To find $T_{k}$, we can now substitute the values of $a$ and $d$ into the formula for the general term and simplify.

We were looking for $b$ and $c$ in the expression $T_{k} = bk + c$. So the correct answers are

1. $b = 6$
2. $c = -9$

You may find it helpful to revise sigma notation in the Everything Maths textbook.

Writing $S_{n}$ in sigma notation, we get:

So the correct answers are

1. $x: k = 1$
2. $y: n$

Substitute the values of $a$ and $d$ into the formula for the sum of the series.

The formula for the sum of $n$ terms in an arithmetic series is:

We need to substitute the values of $a$ and $d$ into this formula, and simplify.

First find the values of $a$ and $d$ for this series.

The formula for the general term of an arithmetic series is

$a$ is the first term of the series. So here $a = 4$.

$d$ is the constant difference between the terms of the series. We can determine this by finding the difference between any two consecutive terms of the series. Taking the difference of the first two terms:

To find $T_{k}$, we can now substitute the values of $a$ and $d$ into the formula for the general term and simplify.

We were looking for $b$ and $c$ in the expression $T_{k} = bk + c$. So the correct answers are

1. $b = -6$
2. $c = 10$

You may find it helpful to revise sigma notation in the Everything Maths textbook.

Writing $S_{n}$ in sigma notation, we get:

So the correct answers are

1. $x: k = 1$
2. $y: n$

Substitute the values of $a$ and $d$ into the formula for the sum of the series.

The formula for the sum of $n$ terms in an arithmetic series is:

We need to substitute the values of $a$ and $d$ into this formula, and simplify.

First find the values of $a$ and $d$ for this series.

The formula for the general term of an arithmetic series is

$a$ is the first term of the series. So here $a = 2$.

$d$ is the constant difference between the terms of the series. We can determine this by finding the difference between any two consecutive terms of the series. Taking the difference of the first two terms:

To find $T_{k}$, we can now substitute the values of $a$ and $d$ into the formula for the general term and simplify.

We were looking for $b$ and $c$ in the expression $T_{k} = bk + c$. So the correct answers are

1. $b = -8$
2. $c = 10$

You may find it helpful to revise sigma notation in the Everything Maths textbook.

Writing $S_{n}$ in sigma notation, we get:

So the correct answers are

1. $x: k = 1$
2. $y: n$

Substitute the values of $a$ and $d$ into the formula for the sum of the series.

The formula for the sum of $n$ terms in an arithmetic series is:

We need to substitute the values of $a$ and $d$ into this formula, and simplify.

Taking a closer look at the terms of $Q$, we can see that each new term of $Q$ is the same as the term before it, with one extra number at the end.

$Q_{1} = -7$ is a random number. But each of the other numbers which are being added to $Q$ is a term of $S_{n}$. For instance:

and

and so on. So the general term for $Q_{n}$ is

$S_{n}$ has a constant difference of $d = 4$. So we can use this to find the missing values of $Q_{6}$.

Use the fact that you know $Q_{n}$ in terms of $S$, and that you are given an algebraic expression for $S_{n}$.

From Question 1, we know that

And we are given in the question that

We can use these together to find the value of $Q_{111}$.

Taking a closer look at the terms of $Q$, we can see that each new term of $Q$ is the same as the term before it, with one extra number at the end.

$Q_{1} = -9$ is a random number. But each of the other numbers which are being added to $Q$ is a term of $S_{n}$. For instance:

and

and so on. So the general term for $Q_{n}$ is

$S_{n}$ has a constant difference of $d = 4$. So we can use this to find the missing values of $Q_{7}$.

Use the fact that you know $Q_{n}$ in terms of $S$, and that you are given an algebraic expression for $S_{n}$.

From Question 1, we know that

And we are given in the question that

We can use these together to find the value of $Q_{121}$.

Taking a closer look at the terms of $Q$, we can see that each new term of $Q$ is the same as the term before it, with one extra number at the end.

$Q_{1} = 7$ is a random number. But each of the other numbers which are being added to $Q$ is a term of $S_{n}$. For instance:

and

and so on. So the general term for $Q_{n}$ is

$S_{n}$ has a constant difference of $d = -8$. So we can use this to find the missing values of $Q_{7}$.

Use the fact that you know $Q_{n}$ in terms of $S$, and that you are given an algebraic expression for $S_{n}$.

From Question 1, we know that

And we are given in the question that

We can use these together to find the value of $Q_{115}$.

You can calculate the sum of the first $n$ terms of an arithmetic series using the formula $S_{n} = \dfrac{n}{2} \left(a + l \right)$.

For this question we have the first and last terms of an arithmetic series, and we need to calculate the sum of the first $\text{17}$ terms in the series.

The formula for the sum of terms in an arithmetic series is $S_{n} = \dfrac{n}{2} \left( a + l \right)$:

• $S_{n}$ is the sum we want
• $n$ is the number of terms we are summing
• $a$ is the first term of the series
• $l$ is the last term of the series

We need the values of $n, a$ and $l$ - then we can substitute them into the formula and evaluate the answer.

We can take the value of $n$ straight from the question: it is in the words "the sum of these $\text{17}$ terms" as well as in the symbol "$S_{17}$." Since we want to add up $17$ terms of the series, $n = 17$.

The first term is given in the question: $a = 10$. We can also get the last term in the series directly from the question: $l = -70$.

We now have the values $n = 17, a = 10$ and $l = -70$.

Now substitute the values into the formula and work out the answer.

The sum of the first $\text{17}$ terms for the series is $S_{17} = -510$.

You can solve this in the same way as the first question. However, there is a quicker way!

We need to determine the sum $S_{18}$. Since the question is only worth one mark, we can expect that there is a quick way (faster than the calculation in the first question).

Compare this question to the first one: in the first question we found the sum $S_{17}$. Think about what this means: it is the sum of the first $\text{17}$ terms, as shown below.

Now we want to find the sum of the first $\text{18}$ terms, which is the same as the sum above, except with one more term added on! In other words, $S_{18} = S_{17} + T_{18}$. We can substitute the values we know into this equation to get the answer.

The sum of the first $\text{18}$ terms for the series is $S_{18} = -585$.

You can calculate the sum of the first $n$ terms of an arithmetic series using the formula $S_{n} = \dfrac{n}{2} \left(a + l \right)$.

For this question we have the first three terms and the last term of an arithmetic series, and we need to calculate the sum of the first $\text{27}$ terms in the series.

The formula for the sum of terms in an arithmetic series is $S_{n} = \dfrac{n}{2} \left( a + l \right)$:

• $S_{n}$ is the sum we want
• $n$ is the number of terms we are summing
• $a$ is the first term of the series
• $l$ is the last term of the series

We need the values of $n, a$ and $l$ - then we can substitute them into the formula and evaluate the answer.

We can take the value of $n$ straight from the question: it is in the words "the sum of these $\text{27}$ terms" as well as in the symbol "$S_{27}$." Since we want to add up $27$ terms of the series, $n = 27$.

The first term is given in the question: $a = -4$. The last term is also given in the question $l = 22$. (The question also shows the second and third terms, but we do not need them to answer the question.)

We now have the values $n = 27, a = -4$ and $l = 22$.

Now substitute the values into the formula and work out the answer.

The sum of the first $\text{27}$ terms for the series is $S_{27} = 243$.

You can solve this in the same way as the first question. However, there is a quicker way!

We need to determine the sum $S_{26}$. Since the question is only worth one mark, we can expect that there is a quick way (faster than the calculation in the first question).

Compare this question to the first one: in the first question we found the sum $S_{27}$. Think about what this means: it is the sum of the first $\text{27}$ terms, as shown below.

Now we want to find the sum of the first $\text{26}$ terms, which is the same as above except with the last term taken away! In other words, $S_{26} = S_{27} - T_{27}$. We can substitute the values we know into this equation to get the answer.

The sum of the first $\text{26}$ terms for the series is $S_{26} = 221$.

You can calculate the sum of the first $n$ terms of an arithmetic series using the formula $S_{n} = \dfrac{n}{2} \left(a + l \right)$.

For this question we have the first and last terms of an arithmetic series, and we need to calculate the sum of the first $\text{23}$ terms in the series.

The formula for the sum of terms in an arithmetic series is $S_{n} = \dfrac{n}{2} \left( a + l \right)$:

• $S_{n}$ is the sum we want
• $n$ is the number of terms we are summing
• $a$ is the first term of the series
• $l$ is the last term of the series

We need the values of $n, a$ and $l$ - then we can substitute them into the formula and evaluate the answer.

We can take the value of $n$ straight from the question: it is in the words "the sum of these $\text{23}$ terms" as well as in the symbol "$S_{23}$." Since we want to add up $23$ terms of the series, $n = 23$.

The first term is given in the question: $a = 9$. We can also get the last term in the series directly from the question: $l = 97$.

We now have the values $n = 23, a = 9$ and $l = 97$.

Now substitute the values into the formula and work out the answer.

The sum of the first $\text{23}$ terms for the series is $S_{23} = 1219$.

You can solve this in the same way as the first question. However, there is a quicker way!

We need to determine the sum $S_{22}$. Since the question is only worth one mark, we can expect that there is a quick way (faster than the calculation in the first question).

Compare this question to the first one: in the first question we found the sum $S_{23}$. Think about what this means: it is the sum of the first $\text{23}$ terms, as shown below.

Now we want to find the sum of the first $\text{22}$ terms, which is the same as above except with the last term taken away! In other words, $S_{22} = S_{23} - T_{23}$. We can substitute the values we know into this equation to get the answer.

The sum of the first $\text{22}$ terms for the series is $S_{22} = 1122$.

Use the equation $S_{n} = \frac{n}{2}(2a + (n - 1)d)$ to write an equation using the information about the sum of the first 38 terms. Then write out another equation based on the difference between the sixth and third terms.

There is a lot of information in the question. We will start with the statement that the sum of the first 38 terms is -5396. This information fits with the formula $S_{n} = \frac{n}{2}(2a + (n - 1)d)$, which connects the sum of $n$ terms of an arithmetic series to the values of $a$ and $d$.

Substitute in what we know to see what the equation can give us:

For the moment we are stuck because there are two variables in the equation ($a$ and $d$), so we need to let this equation sit for the moment while we see what else we can get out of the question.

In the work above, we got an equation with two variables, so the plan for the question is now clear: find another equation with the variables $a$ and $d$ so that we can solve the equations simultaneously. Let us try to do that using the information in the question about the other terms.

In this case, the question states that the "difference between the third and sixth terms ... is -24". We can write this as an equation: $T_{6} - T_{3} = -24$. Using the relationship $T_{n} = a + (n - 1)d$, we can write this equation in terms of $a$ and $d$:

This is fantastic: the $a$'s cancelled so we could actually find the value of the common difference, $d$.

We now have the two equations:

Substitute $d = -8$ into equation ($\text{1}$) to find the value of $a$.

We now have the value of the first term, $T_{1} = a = 6$, and the common difference, $d = -8$.

Now use the values of $a = 6$ and $d = -8$ to calculate the values of $T_{2}$ and $T_{3}$.

The first three terms of the sequence are: $T_{1} = 6, T_{2} = -2, T_{3} = -10$.

You can find the answer by comparing the sums $S_{39}$ and $S_{38}$.

$S_{n}$ is the sum of $n$ terms of a series: $S_{n} = T_{1} + T_{2} + T_{3} + \ldots + T_{n}$. In this question we have $S_{39}$ and $S_{38}$:

As you can see, the only difference between $S_{39}$ and $S_{38}$ is one term, $T_{39}$. If we subtract $T_{39}$ from $S_{39}$ we will get $S_{38}$. In other words, $T_{39}$ is the difference between $S_{39}$ and $S_{38}$:

Now we can substitute in the values of the two sums ($S_{39} = -5694$ and $S_{38} = -5396$) and find the term we want:

The correct answer is $T_{39} = -298$.

Use the equation $S_{n} = \frac{n}{2}(2a + (n - 1)d)$ to write an equation using the information about the sum of the first 28 terms. Then write out another equation based on the either $T_{4}$ or $T_{9}$.

There is a lot of information in the question. We will start with the statement that the sum of the first 28 terms is -1638. This information fits with the formula $S_{n} = \frac{n}{2}(2a + (n - 1)d)$, which connects the sum of $n$ terms of an arithmetic series to the values of $a$ and $d$.

Substitute in what we know to see what the equation can give us:

For the moment we are stuck because there are two variables in the equation ($a$ and $d$), so we need to let this equation sit for the moment while we see what else we can get out of the question.

In the work above, we got an equation with two variables, so the plan for the question is now clear: find another equation with the variables $a$ and $d$ so that we can solve the equations simultaneously. Let us try to do that using the information in the question about the other terms.

In this case, we have the values for two different terms, $T_{4}$ and $T_{9}$. We can use either one of them to write an equation using the formula $T_{n} = a + (n - 1)d$. We will use $T_{4} = -6$.

We now have the two equations:

Now we can solve the equations simultaneously. Multiply equation ($\text{2}$) by $\text{2}$ to get $2a + 6d = -12$: that way both equations include $2a$ and we can use elimination.

We now have the value of the first term, $T_{1} = a = 9$, and the common difference, $d = -5$.

Now use the values of $a = 9$ and $d = -5$ to calculate the values of $T_{2}$ and $T_{3}$.

The first three terms of the sequence are: $T_{1} = 9, T_{2} = 4, T_{3} = -1$.

You can find the answer by comparing the sums $S_{28}$ and $S_{27}$.

$S_{n}$ is the sum of $n$ terms of a series: $S_{n} = T_{1} + T_{2} + T_{3} + \ldots + T_{n}$. In this question we have $S_{28}$ and $S_{27}$:

As you can see, the only difference between $S_{28}$ and $S_{27}$ is one term, $T_{28}$. If we subtract $T_{28}$ from $S_{28}$ we will get $S_{27}$. In other words, $T_{28}$ is the difference between $S_{28}$ and $S_{27}$:

Now we can substitute in the values of the two sums ($S_{28} = -1638$ and $S_{27} = -1512$) and find the term we want:

The correct answer is $T_{28} = -126$.

Use the equation $S_{n} = \frac{n}{2}(2a + (n - 1)d)$ to write an equation using the information about the sum of the first 28 terms. Then write out another equation based on the difference between the fourth and second terms.

There is a lot of information in the question. We will start with the statement that the sum of the first 28 terms is -1022. This information fits with the formula $S_{n} = \frac{n}{2}(2a + (n - 1)d)$, which connects the sum of $n$ terms of an arithmetic series to the values of $a$ and $d$.

Substitute in what we know to see what the equation can give us:

For the moment we are stuck because there are two variables in the equation ($a$ and $d$), so we need to let this equation sit for the moment while we see what else we can get out of the question.

In the work above, we got an equation with two variables, so the plan for the question is now clear: find another equation with the variables $a$ and $d$ so that we can solve the equations simultaneously. Let us try to do that using the information in the question about the other terms.

In this case, the question states that the "difference between the second and fourth terms ... is -6". We can write this as an equation: $T_{4} - T_{2} = -6$. Using the relationship $T_{n} = a + (n - 1)d$, we can write this equation in terms of $a$ and $d$:

This is fantastic: the $a$'s cancelled so we could actually find the value of the common difference, $d$.

We now have the two equations:

Substitute $d = -3$ into equation ($\text{1}$) to find the value of $a$.

We now have the value of the first term, $T_{1} = a = 4$, and the common difference, $d = -3$.

Now use the values of $a = 4$ and $d = -3$ to calculate the values of $T_{2}$ and $T_{3}$.

The first three terms of the sequence are: $T_{1} = 4, T_{2} = 1, T_{3} = -2$.

You can find the answer by comparing the sums $S_{29}$ and $S_{28}$.

$S_{n}$ is the sum of $n$ terms of a series: $S_{n} = T_{1} + T_{2} + T_{3} + \ldots + T_{n}$. In this question we have $S_{29}$ and $S_{28}$:

As you can see, the only difference between $S_{29}$ and $S_{28}$ is one term, $T_{29}$. If we subtract $T_{29}$ from $S_{29}$ we will get $S_{28}$. In other words, $T_{29}$ is the difference between $S_{29}$ and $S_{28}$:

Now we can substitute in the values of the two sums ($S_{29} = -1102$ and $S_{28} = -1022$) and find the term we want:

The correct answer is $T_{29} = -80$.

You can calculate the sum of the first $n$ terms of an arithmetic series using the formula $S_{n} = \dfrac{n}{2} \left( 2a + (n-1)d \right)$.

The question gives us the first three terms of an arithmetic series, and we need to calculate the sum of the first $\text{12}$ terms in the series.

The formula for the sum of terms in an arithmetic series is $S_{n} = \dfrac{n}{2} \left( 2a + (n-1)d \right)$:

• $S_{n}$ is the sum we want
• $n$ is the number of terms we are summing
• $a$ is the first term of the series
• $d$ is the common difference in the series

We need the values of $n, a$ and $d$ - then we can substitute them into the formula and evaluate the answer.

The value of $n$ is sitting in the question: we can find it in the words "the sum of the first $\text{12}$ terms" as well as in the symbol "$S_{\text{12}}$." For this question, we want to add up $\text{12}$ terms of the series, so $n = \text{12}$.

The first term is given in the question: $a = 4$. The common difference is not given so clearly in the question - we must calculate it by using the terms given: $d = T_{2} - T_{1} = 9 - 4 = 5$.

We now have the values $n = 12, a = 4$ and $d = 5$.

Now substitute the values into the formula and work out the answer.

The sum of the first $\text{12}$ terms for the series is $S_{12} = 378$.

You can solve this in the same way as the first question. However, there is a quicker way!

We now have the new information $T_{12} = 59$ and need to determine the sum $S_{11}$. One option is to use the formula again, as in the first question. However, there is a much quicker way (and the question is only worth one mark).

Compare this question to the first one: in the first question we found the sum of the first $\text{12}$ terms, as shown below.

Now we want to find the sum of the first $\text{11}$ terms, which is the same as above except with the last term taken away! In other words, $S_{11} = S_{12} - T_{12}$. We can substitute the values we know into this equation to get the answer.

The sum of the first $\text{11}$ terms for the series is $S_{11} = 319$.

You can calculate the sum of the first $n$ terms of an arithmetic series using the formula $S_{n} = \dfrac{n}{2} \left( 2a + (n-1)d \right)$.

The question gives us the first and common difference of an arithmetic series, and we need to calculate the sum of the first $\text{18}$ terms in the series.

The formula for the sum of terms in an arithmetic series is $S_{n} = \dfrac{n}{2} \left( 2a + (n-1)d \right)$:

• $S_{n}$ is the sum we want
• $n$ is the number of terms we are summing
• $a$ is the first term of the series
• $d$ is the common difference in the series

We need the values of $n, a$ and $d$ - then we can substitute them into the formula and evaluate the answer.

The value of $n$ is sitting in the question: we can find it in the words "the sum of the first $\text{18}$ terms" as well as in the symbol "$S_{\text{18}}$." For this question, we want to add up $\text{18}$ terms of the series, so $n = \text{18}$.

The first term is given in the question: $a = -8$. The common difference is also clearly stated in the question: $d = 3$.

We now have the values $n = 18, a = -8$ and $d = 3$.

Now substitute the values into the formula and work out the answer.

The sum of the first $\text{18}$ terms for the series is $S_{18} = 315$.

You can solve this in the same way as the first question. However, there is a quicker way!

We now have the new information $T_{18} = 43$ and need to determine the sum $S_{17}$. One option is to use the formula again, as in the first question. However, there is a much quicker way (and the question is only worth one mark).

Compare this question to the first one: in the first question we found the sum of the first $\text{18}$ terms, as shown below.

Now we want to find the sum of the first $\text{17}$ terms, which is the same as above except with the last term taken away! In other words, $S_{17} = S_{18} - T_{18}$. We can substitute the values we know into this equation to get the answer.

The sum of the first $\text{17}$ terms for the series is $S_{17} = 272$.

You can calculate the sum of the first $n$ terms of an arithmetic series using the formula $S_{n} = \dfrac{n}{2} \left( 2a + (n-1)d \right)$.

The question gives us the first three terms of an arithmetic series, and we need to calculate the sum of the first $\text{16}$ terms in the series.

The formula for the sum of terms in an arithmetic series is $S_{n} = \dfrac{n}{2} \left( 2a + (n-1)d \right)$:

• $S_{n}$ is the sum we want
• $n$ is the number of terms we are summing
• $a$ is the first term of the series
• $d$ is the common difference in the series

We need the values of $n, a$ and $d$ - then we can substitute them into the formula and evaluate the answer.

The value of $n$ is sitting in the question: we can find it in the words "the sum of the first $\text{16}$ terms" as well as in the symbol "$S_{\text{16}}$." For this question, we want to add up $\text{16}$ terms of the series, so $n = \text{16}$.

The first term is given in the question: $a = -7$. The common difference is not given so clearly in the question - we must calculate it by using the terms given: $d = T_{2} - T_{1} = (-10) - (-7) = -3$.

We now have the values $n = 16, a = -7$ and $d = -3$.

Now substitute the values into the formula and work out the answer.

The sum of the first $\text{16}$ terms for the series is $S_{16} = -472$.

You can solve this in the same way as the first question. However, there is a quicker way!

We now have the new information $T_{16} = -52$ and need to determine the sum $S_{15}$. One option is to use the formula again, as in the first question. However, there is a much quicker way (and the question is only worth one mark).

Compare this question to the first one: in the first question we found the sum of the first $\text{16}$ terms, as shown below.

Now we want to find the sum of the first $\text{15}$ terms, which is the same as above except with the last term taken away! In other words, $S_{15} = S_{16} - T_{16}$. We can substitute the values we know into this equation to get the answer.

The sum of the first $\text{15}$ terms for the series is $S_{15} = -420$.

You may find it helpful to revise arithmetic sequences in the Everything Maths textbook.

We can see from the first three terms that the common difference value for this sequence is $\text{3}$. So the sixth term can be calculated as:

You may find it helpful to revise arithmetic sequences in the Everything Maths textbook.

We can use the same formula that we used in Question 1 to find the number of terms in the sequence, because we know the value of the final term in the sequence:

You may find it helpful to revise finite arithmetic series in the Everything Maths textbook.

The first two terms are negative, and then all of the rest of the terms are positive. So we need to exclude the first two terms when we determine the sum of the positive terms.

So the positive terms of the sequence look like:

There will be $\text{19}$ terms in this sequence, since we are excluding two negative terms. And the new value of $a$ will be $\text{2}$. We can then calculate the sum of these terms using the formula for the sum of an arithmetic series:

You may find it helpful to revise finite arithmetic series in the Everything Maths textbook.

The first term in the sequence is $-\text{4}$, which is divisible by $\text{4}$.

But the last term in the sequence is $\text{2,873}$, which is not divisible by $\text{4}$. So we need to work out the value of the last term which is divisible by $\text{4}$.

Since the common difference of the sequence is $\text{3}$, we can count back from $\text{2,873}$ in steps of $-\text{3}$ until we reach a number which is divisible by $\text{4}$.

So the last term which is divisible by $\text{4}$ is $\text{2,864}$.

The original sequence is:

The numbers divisible by $\color{red}{\text{4}}$ form a new sequence: $-\text{4}\, ,\text{8}\, ,\text{20}\, ,\ldots \, , \text{2,864}$

The new pattern is an arithmetic sequence with $a = -\text{4}$ and $d = \text{12}$.

You may find it helpful to revise arithmetic sequences in the Everything Maths textbook.

We can see from the first three terms that the common difference value for this sequence is $-\text{4}$. So the fifth term can be calculated as:

You may find it helpful to revise arithmetic sequences in the Everything Maths textbook.

We can use the same formula that we used in Question 1 to find the number of terms in the sequence, because we know the value of the final term in the sequence:

You may find it helpful to revise finite arithmetic series in the Everything Maths textbook.

The first two terms are positive, and then all of the rest of the terms are negative. So we need to exclude the first two terms when we determine the sum of the negative terms.

So the negative terms of the sequence look like:

There will be $\text{24}$ terms in this sequence, since we are excluding two positive terms. And the new value of $a$ will be $-\text{3}$. We can then calculate the sum of these terms using the formula for the sum of an arithmetic series:

You may find it helpful to revise finite arithmetic series in the Everything Maths textbook.

The first term in the sequence is $\text{5}$, which is divisible by $\text{5}$.

But the last term in the sequence is $-\text{4,991}$, which is not divisible by $\text{5}$. So we need to work out the value of the last term which is divisible by $\text{5}$.

Since the common difference of the sequence is $-\text{4}$, we can count back from $-\text{4,991}$ in steps of $\text{4}$ until we reach a number which is divisible by $\text{5}$.

So the last term which is divisible by $\text{5}$ is $-\text{4,975}$.

The original sequence is:

The numbers divisible by $\color{red}{\text{5}}$ form a new sequence: $\text{5}\, ,-\text{15}\, ,-\text{35}\, ,\ldots \, , -\text{4,975}$

The new pattern is an arithmetic sequence with $a = \text{5}$ and $d = -\text{20}$.

You may find it helpful to revise arithmetic sequences in the Everything Maths textbook.

We can see from the first three terms that the common difference value for this sequence is $\text{4}$. So the fourth term can be calculated as:

You may find it helpful to revise arithmetic sequences in the Everything Maths textbook.

We can use the same formula that we used in Question 1 to find the number of terms in the sequence, because we know the value of the final term in the sequence:

You may find it helpful to revise finite arithmetic series in the Everything Maths textbook.

The first two terms are negative, and then all of the rest of the terms are positive. So we need to exclude the first two terms when we determine the sum of the positive terms.

So the positive terms of the sequence look like:

There will be $\text{24}$ terms in this sequence, since we are excluding two negative terms. And the new value of $a$ will be $\text{2}$. We can then calculate the sum of these terms using the formula for the sum of an arithmetic series:

You may find it helpful to revise finite arithmetic series in the Everything Maths textbook.

The first term in the sequence is $-\text{6}$, which is divisible by $\text{3}$.

But the last term in the sequence is $\text{2,390}$, which is not divisible by $\text{3}$. So we need to work out the value of the last term which is divisible by $\text{3}$.

Since the common difference of the sequence is $\text{4}$, we can count back from $\text{2,390}$ in steps of $-\text{4}$ until we reach a number which is divisible by $\text{3}$.

So the last term which is divisible by $\text{3}$ is $\text{2,382}$.

The original sequence is:

The numbers divisible by $\color{red}{\text{3}}$ form a new sequence: $-\text{6}\, ,\text{6}\, ,\text{18}\, ,\ldots \, , \text{2,382}$

The new pattern is an arithmetic sequence with $a = -\text{6}$ and $d = \text{12}$.

Start by finding the values of $a$ and $d$. Then use the formula $T_{n} = a + (n - 1)d$.

The question gives us an arithmetic series and we must determine the number of terms in the series. We can do this using the formula $T_{n} = a + (n - 1)d$.

The number of terms in the series is represented by the $n$-value of the last term in the series. In this case, the last term is $-331$. We can then substitute into the formula as follows:

If we also substitute in the values of the first term, $a$, and the common difference, $d$, we can solve the equation for the value of $n$ for the last term. This will give us the total number of terms in the series.

From the series in the question, we can get both the first term and the common difference. The first term is sitting there for us to take: $a = 5$. The common difference requires a bit of work: it is the difference between any two consecutive terms in the series: $d = T_{2} - T_{1}$, so we get $d = 1 - 5 = -4$.

Substitute these values into the equation from above:

Now we can solve the equation for $n_{\small\text{last}}$. Start by distributing the $-\text{4}$ into the brackets.

The $n$-value of the final term is $\text{85}$.

The calculation here shows that there are $\text{85}$ terms in the series.

You can use the same method as in the first question to find the answer: substitute values in the equation $T_{n} = a + (n - 1)d$ and find the value of $n$. The answer you get for $n$ will tell you if $-\text{197}$ belongs in the series or not.

This question asks us to figure out whether or not $-\text{197}$ is a number in the series. In other words, if we write out every term in the series $5 +1 -3 - \ldots -327 -331$, will we find $-\text{197}$ somewhere in the middle or will the numbers "jump" over it?

We can work out the answer using the same method as the first question - by finding the value of $n$ for the number $-\text{197}$. Basically, we will assume that $-\text{197}$ is a term in the series and then use the value of $n$ to check whether or not it is true! Substitute the values into the equation and calculate $n$.

This calculation shows that if the term is $-\text{197}$ then $n = \text{51.5}$.

How does this calculation show us whether $-\text{197}$ is a term in the series or not? It depends on whether or not the value for $n$ makes sense. In this case, we found that $n \approx \text{51.5}$, which does not make sense because it is a decimal number (not a whole number). $n$ represents the position of the term in the series. Finding a term in position $n \approx \text{51.5}$ is the same as saying that you stood in a queue of $\text{85}$ people and you were $\text{51.5}$^th person in the queue. That is impossible!

The correct answer is no, the number is not a term in the series because the $n$-value is not a natural number.

Start by finding the values of $a$ and $d$. Then use the formula $T_{n} = a + (n - 1)d$.

The question gives us an arithmetic series and we must determine the number of terms in the series. We can do this using the formula $T_{n} = a + (n - 1)d$.

The number of terms in the series is represented by the $n$-value of the last term in the series. In this case, the last term is $-3$. We can then substitute into the formula as follows:

If we also substitute in the values of the first term, $a$, and the common difference, $d$, we can solve the equation for the value of $n$ for the last term. This will give us the total number of terms in the series.

From the series in the question, we can get both the first term and the common difference. The first term is sitting there for us to take: $a = 240$. The common difference requires a bit of work: it is the difference between any two consecutive terms in the series: $d = T_{2} - T_{1}$, so we get $d = 237 - 240 = -3$.

Substitute these values into the equation from above:

Now we can solve the equation for $n_{\small\text{last}}$. Start by distributing the $-\text{3}$ into the brackets.

The $n$-value of the final term is $\text{82}$.

The calculation here shows that there are $\text{82}$ terms in the series.

You can use the same method as in the first question to find the answer: substitute values in the equation $T_{n} = a + (n - 1)d$ and find the value of $n$. The answer you get for $n$ will tell you if $\text{135}$ belongs in the series or not.

This question asks us to figure out whether or not $\text{135}$ is a number in the series. In other words, if we write out every term in the series $240 +237 +234 + \ldots +0 -3$, will we find $\text{135}$ somewhere in the middle or will the numbers "jump" over it?

We can work out the answer using the same method as the first question - by finding the value of $n$ for the number $\text{135}$. Basically, we will assume that $\text{135}$ is a term in the series and then use the value of $n$ to check whether or not it is true! Substitute the values into the equation and calculate $n$.

This calculation shows that if the term is $\text{135}$ then $n = \text{36}$.

How does this calculation show us whether $\text{135}$ is a term in the series or not? It depends on whether or not the value for $n$ makes sense. The answer $n = \text{36}$ does make sense because it is a natural number between $\text{1}$ and $\text{82}$. $n = \text{36}$ means that $\text{135}$ is the $\text{36}$^th term in the series. (For comparison, suppose we got the answer $n = -\text{36}$ or $n = \text{83}$; those answers are impossible, because we already know that the series starts at the first term, $n = 1$, and ends at $n = 82$.)

The correct answer is yes, the number is a term in the series because the $n$-value is a natural number.

Start by finding the values of $a$ and $d$. Then use the formula $T_{n} = a + (n - 1)d$.

The question gives us an arithmetic series and we must determine the number of terms in the series. We can do this using the formula $T_{n} = a + (n - 1)d$.

The number of terms in the series is represented by the $n$-value of the last term in the series. In this case, the last term is $42$. We can then substitute into the formula as follows:

If we also substitute in the values of the first term, $a$, and the common difference, $d$, we can solve the equation for the value of $n$ for the last term. This will give us the total number of terms in the series.

From the series in the question, we can get both the first term and the common difference. The first term is sitting there for us to take: $a = -1$. The common difference requires a bit of work: it is the difference between any two consecutive terms in the series: $d = T_{2} - T_{1}$, so we get $d = 0 - (-1) = 1$.

Substitute these values into the equation from above:

Now we can solve the equation for $n_{\small\text{last}}$. Start by distributing the $\text{1}$ into the brackets.

The $n$-value of the final term is $\text{44}$.

The calculation here shows that there are $\text{44}$ terms in the series.

You can use the same method as in the first question to find the answer: substitute values in the equation $T_{n} = a + (n - 1)d$ and find the value of $n$. The answer you get for $n$ will tell you if $\text{37}$ belongs in the series or not.

This question asks us to figure out whether or not $\text{37}$ is a number in the series. In other words, if we write out every term in the series $-1 +0 +1 + \ldots +41 +42$, will we find $\text{37}$ somewhere in the middle or will the numbers "jump" over it?

For the series $-1 +0 +1 + \ldots +41 +42$, it is clear that the number $\text{37}$ is in the series. However, we will show that it is true with a calculation, as the question instructs.

We can work out the answer using the same method as the first question - by finding the value of $n$ for the number $\text{37}$. Basically, we will assume that $\text{37}$ is a term in the series and then use the value of $n$ to check whether or not it is true! Substitute the values into the equation and calculate $n$.

This calculation shows that if the term is $\text{37}$ then $n = \text{39}$.

How does this calculation show us whether $\text{37}$ is a term in the series or not? It depends on whether or not the value for $n$ makes sense. The answer $n = \text{39}$ does make sense because it is a natural number between $\text{1}$ and $\text{44}$. $n = \text{39}$ means that $\text{37}$ is the $\text{39}$^th term in the series. (For comparison, suppose we got the answer $n = -\text{39}$ or $n = \text{45}$; those answers are impossible, because we already know that the series starts at the first term, $n = 1$, and ends at $n = 44$.)

The correct answer is yes, the number is a term in the series because the $n$-value is a natural number.

The constant ratio, $r$, is the quotient of one term and the term before it. For example,

We have the geometric sequence $\frac{1}{2};-1;2; \ldots$ and we need to figure out the constant ratio as well as the general formula. The constant ratio is the number (or expression) we need to multiply each term by to get the next term. We can calculate the ratio by finding the quotient of any two consecutive terms: $r = \frac{T_3}{T_2}$ or $\frac{T_2}{T_1}$ or $\frac{T_{n}}{T_{n-1}}$. Therefore, we can pick any pair of consecutive terms from the sequence. This time let's use the second and third terms:

Notice that the ratio is negative: that is the reason why the terms in the sequence keep changing signs. (Multiplying by a negative always changes the sign.)

Now that we have calculated the value for $r$, we can calculate the general formula for $\frac{1}{2};-1;2; \ldots$. To find the general formula for a geometric sequence, we use the equation $T_n=ar^{n-1}$. The variable $a$ is the first term in the sequence and $r$ is the constant ratio.

The question states that the first three terms of the sequence are $\frac{1}{2};-1;2$. So, the value for $a$ is $\frac{1}{2}$. Substituting $a = \frac{1}{2}$ and $r = -2$ into the general formula, we get:

Do not let the variables in the sequence bother you: this question can be solved in the same way as the first question.

As with the first question, we have a geometric sequence, $-3;3 x;- 3 x^{2}; \ldots$, and we need to figure out the constant ratio as well as the general formula. We calculate the ratio for this sequence in the same way as for the previous question: use any two consecutive terms: $r = \frac{T_3}{T_2}$ or $\frac{T_2}{T_1}$ or $\frac{T_{n-1}}{T_n}$. Since we can choose which terms we use, pick the simplest ones available for the calculation. Then simplify.

Now we will find the general formula for the sequence. This follows the exact same process as in the first question: we use $T_n=ar^{n-1}$ and substitute in the correct values for $a$ and $r$ for this sequence. The first term of the sequence is $a = -3$ while the constant ratio is $r = - x$. Substitute these values into the formula:

The constant ratio, $r$, is the quotient of one term and the term before it. For example,

We have the geometric sequence $-7;21;-63; \ldots$ and we need to figure out the constant ratio as well as the general formula. The constant ratio is the number (or expression) we need to multiply each term by to get the next term. We can calculate the ratio by finding the quotient of any two consecutive terms: $r = \frac{T_3}{T_2}$ or $\frac{T_2}{T_1}$ or $\frac{T_{n}}{T_{n-1}}$. Therefore, we can pick any pair of consecutive terms from the sequence. This time let's use the first and second terms:

Notice that the ratio is negative: that is the reason why the terms in the sequence keep changing signs. (Multiplying by a negative always changes the sign.)

Now that we have calculated the value for $r$, we can calculate the general formula for $-7;21;-63; \ldots$. To find the general formula for a geometric sequence, we use the equation $T_n=ar^{n-1}$. The variable $a$ is the first term in the sequence and $r$ is the constant ratio.

The question states that the first three terms of the sequence are $-7;21;-63$. So, the value for $a$ is $-7$. Substituting $a = -7$ and $r = -3$ into the general formula, we get:

Do not let the variables in the sequence bother you: this question can be solved in the same way as the first question.

As with the first question, we have a geometric sequence, $-5;5 x^{2};- 5 x^{4}; \ldots$, and we need to figure out the constant ratio as well as the general formula. We calculate the ratio for this sequence in the same way as for the previous question: use any two consecutive terms: $r = \frac{T_3}{T_2}$ or $\frac{T_2}{T_1}$ or $\frac{T_{n-1}}{T_n}$. Since we can choose which terms we use, pick the simplest ones available for the calculation. Then simplify.

Now we will find the general formula for the sequence. This follows the exact same process as in the first question: we use $T_n=ar^{n-1}$ and substitute in the correct values for $a$ and $r$ for this sequence. The first term of the sequence is $a = -5$ while the constant ratio is $r = - x^{2}$. Substitute these values into the formula:

The constant ratio, $r$, is the quotient of one term and the term before it. For example,

We have the geometric sequence $\frac{1}{3};-1;3; \ldots$ and we need to figure out the constant ratio as well as the general formula. The constant ratio is the number (or expression) we need to multiply each term by to get the next term. We can calculate the ratio by finding the quotient of any two consecutive terms: $r = \frac{T_3}{T_2}$ or $\frac{T_2}{T_1}$ or $\frac{T_{n}}{T_{n-1}}$. Therefore, we can pick any pair of consecutive terms from the sequence. This time let's use the second and third terms:

Notice that the ratio is negative: that is the reason why the terms in the sequence keep changing signs. (Multiplying by a negative always changes the sign.)

Now that we have calculated the value for $r$, we can calculate the general formula for $\frac{1}{3};-1;3; \ldots$. To find the general formula for a geometric sequence, we use the equation $T_n=ar^{n-1}$. The variable $a$ is the first term in the sequence and $r$ is the constant ratio.

The question states that the first three terms of the sequence are $\frac{1}{3};-1;3$. So, the value for $a$ is $\frac{1}{3}$. Substituting $a = \frac{1}{3}$ and $r = -3$ into the general formula, we get:

Do not let the variables in the sequence bother you: this question can be solved in the same way as the first question.

As with the first question, we have a geometric sequence, $4 x;8 x;16 x; \ldots$, and we need to figure out the constant ratio as well as the general formula. We calculate the ratio for this sequence in the same way as for the previous question: use any two consecutive terms: $r = \frac{T_3}{T_2}$ or $\frac{T_2}{T_1}$ or $\frac{T_{n-1}}{T_n}$. Since we can choose which terms we use, pick the simplest ones available for the calculation. Then simplify.

Now we will find the general formula for the sequence. This follows the exact same process as in the first question: we use $T_n=ar^{n-1}$ and substitute in the correct values for $a$ and $r$ for this sequence. The first term of the sequence is $a = 4 x$ while the constant ratio is $r = 2$. Substitute these values into the formula:

The question asks us to find the geometric mean for the numbers $\frac{1}{2}$ and $\frac{1}{8}$.

This question is the same as: 'Find the value of $y$ so that the terms $\frac{1}{2} \; ; \; y \; ; \; \frac{1}{8}$ follow a geometric sequence.' This is because if $\frac{1}{2} \; ; \; y \; ; \; \frac{1}{8}$ form a geometric sequence, then $y$ is the geometric mean of $\frac{1}{2}$ and $\frac{1}{8}$. (If you have three consecutive terms of a geometric sequence, the middle term is always the geometric mean of the first and third terms.)

Since we are dealing with a geometric sequence, we can find the answer using the constant ratio. Remember that the constant ratio is the quotient of any two consecutive terms in the sequence. We do not know the value of the constant ratio, but we do know that $r$ is equal to both $\frac{T_{2}}{T_{1}}$ and $\frac{T_{3}}{T_{2}}$. So we can write the equation below with the first three terms and then substitute in the terms in this sequence:

The equation above summarizes the relationship between the terms based on the constant ratio.

Now we can solve the equation for $y$.

We get two answers because of the $\pm$ which comes into the solution with the square-root step. This makes sense, because it is possible for the terms of a geometric sequence to change signs: if the constant ratio is positive, the sequence will be $\frac{1}{2} \; ; \; \frac{1}{4} \; ; \;\frac{1}{8}$ while if the constant ratio is negative the sequence will be $\frac{1}{2} \; ; \; - \frac{1}{4} \; ; \;\frac{1}{8}$.

There are two answers at this point, but the question stated that the mean is negative. So we can ignore the $\frac{1}{4}$ answer.

The correct answer is $- \dfrac{1}{4}$.

We need to find the common ratio for the geometric sequence. We know from Question 1 that the sequence is $\frac{1}{2}\; ; \;- \frac{1}{4}\; ; \;\frac{1}{8}$.

The constant ratio is the number (or expression) we need to multiply each term by to get the next term. We can use any two consecutive terms in the sequence to find the answer:

We can summarise this relationship as $r = \frac{T_n}{T_{n-1}}$. In this solution we will use the second and third terms.

Now we can see why the terms in the sequence alternate signs: the constant ratio is negative.

The constant ratio is $r = - \dfrac{1}{2}$.

The question asks us to find the geometric mean for the numbers $-9$ and $-225$.

This question is the same as: 'Find the value of $x$ so that the terms $-9 \; ; \; x \; ; \; -225$ follow a geometric sequence.' This is because if $-9 \; ; \; x \; ; \; -225$ form a geometric sequence, then $x$ is the geometric mean of $-9$ and $-225$. (If you have three consecutive terms of a geometric sequence, the middle term is always the geometric mean of the first and third terms.)

Since we are dealing with a geometric sequence, we can find the answer using the constant ratio. Remember that the constant ratio is the quotient of any two successive terms in the sequence. We do not know the value of the constant ratio, but we do know that $r$ is equal to both $\frac{T_{2}}{T_{1}}$ and $\frac{T_{3}}{T_{2}}$. So we can write the equation below with the first three terms and then substitute in the terms in this sequence:

The equation above summarizes the relationship between the terms based on the constant ratio.

Now we can solve the equation for $x$.

We get two answers because of the $\pm$ which comes into the solution with the square-root step. This makes sense, because it is possible for the terms of a geometric sequence to change signs: if the constant ratio is positive, the sequence will be $-9 \; ; \; -45 \; ; \;-225$ while if the constant ratio is negative the sequence will be $-9 \; ; \; 45 \; ; \;-225$.

There are two answers at this point, but the question stated that the mean is negative. So we can ignore the $45$ answer.

The correct answer is $-45$.

The question asks us to find the general formula for this sequence. The general formula is an expression which tells us how to calculate any term in the sequence.

Start with the equation:

$a$ represents the first term in the sequence, $r$ represents the constant ratio, and $n$ represents the rank (position) of the term in the sequence. For the sequence $-9\; ; \;-45\; ; \;-225$, the first term is $a = -9$. The question tells us that the constant ratio is $r = 5$. Substitute these numbers into the equation and simplify.

The general formula for the sequence is $T_{n} = - \dfrac{9}{5} \left(5\right)^{n}$.

The question asks us to find the geometric mean for the numbers $5$ and $45$.

This question is the same as: 'Find the value of $x$ so that the terms $5 \; ; \; x \; ; \; 45$ follow a geometric sequence.' This is because if $5 \; ; \; x \; ; \; 45$ form a geometric sequence, then $x$ is the geometric mean of $5$ and $45$. (If you have three consecutive terms of a geometric sequence, the middle term is always the geometric mean of the first and third terms.)

Since we are dealing with a geometric sequence, we can find the answer using the constant ratio. Remember that the constant ratio is the quotient of any two consecutive terms in the sequence. We do not know the value of the constant ratio, but we do know that $r$ is equal to both $\frac{T_{2}}{T_{1}}$ and $\frac{T_{3}}{T_{2}}$. So we can write the equation below with the first three terms and then substitute in the terms in this sequence:

The equation above summarizes the relationship between the terms based on the constant ratio.

Now we can solve the equation for $x$.

We get two answers because of the $\pm$ which comes into the solution with the square-root step. This makes sense, because it is possible for the terms of a geometric sequence to change signs: if the constant ratio is positive, the sequence will be $5 \; ; \; 15 \; ; \;45$ while if the constant ratio is negative the sequence will be $5 \; ; \; -15 \; ; \;45$.

There are two answers at this point, but the question stated that the mean is negative. So we can ignore the $15$ answer.

The correct answer is $-15$.

We need to find the common ratio for the geometric sequence. We know from Question 1 that the sequence is $5\; ; \;-15\; ; \;45$.

The constant ratio is the number (or expression) we need to multiply each term by to get the next term. We can use any two consecutive terms in the sequence to find the answer:

We can summarise this relationship as $r = \frac{T_n}{T_{n-1}}$. In this solution we will use the second and third terms.

Now we can see why the terms in the sequence alternate signs: the constant ratio is negative.

The constant ratio is $r = -3$.